CHAPTER VI 
NON-LINEAR SERIES OF GROUPS OF 
POINTS ON A CURVE 
§ 1. General theorems about series 
The systems of groups of points on a base curve which we have 
studied, so far, have all been linear systems. There are, ob 
viously, other types of algebraic series, and it is our purpose in 
the present chapter to find some theorems about them. 
A group of points on a curve will not, necessarily, be the total 
intersection with another curve. A necessary condition for this 
situation is that the number of points on the curve, when 
multiplicities are rightly counted, be divisible by the order of 
the curve. The number of points in a generic group shall still 
be called its order, the number of parameters its ‘dimension’; 
a series of order N and dimension r shall be written as y r N \ we 
shall use the notation g r N only when the series is linear. The 
number of groups to which r generic points of the base curve 
belong is called the ‘index’, a number which is unity for a linear 
series. If every group which contains a generic point P x neces 
sarily goes through ¡x—1 other points P 2 ,P 3 ,..., P^ variable with 
the first, then every group which contains P 2 must also contain 
P v as otherwise P l5 P 2 would impose two conditions, not one. 
Such a series shall be called, as before, an ‘involution’. When 
/x. = 1 the given series is said to be ‘simple’. If the equation of 
the base curve be A 
we may imagine the groups of the series cut by the variable 
curves 
M x >y) = <f>ii x ’y)=- = 0 
00 = 01 
... = 0. 
The equations ifj 0 = 0, i/q = 0 are supposed to involve the 
otherwise arbitrary coefficients of the curves </>. If our series be 
irreducible, we may suppose each of the polynomials indicated 
is irreducible. We may assume that no two curves <f> have the 
same fixed infinite points, and that the axes are so directed 
that, usually, no two points of the same group have the same